Tunable quantum temperature oscillations in graphene nanostructures

We investigate the local electron temperature distribution in graphene nanoribbon and graphene junctions subject to an applied thermal gradient. Using a realistic model of a scanning thermal microscope, we predict quantum temperature oscillations whose wavelength is related to that of Friedel oscillations. Experimentally this wavelength can be tuned over several orders of magnitude by gating or doping, bringing quantum temperature oscillations within reach of the spatial resolution of existing measurement techniques.

Figure 1

A schematic representation of a three terminal GNR junction with the hot and cold electrodes covalently bonded to the GNR and a third scanning thermal probe positioned over the GNR. The probe is allowed to come into thermal and electrical equilibrium with the sample and measure the temperature $T_p$.

Figure 2

The calculated spatial temperature profile for a zigzag GNR probed by a Pt SThM fixed 2.5 Å above the sheet shown for two energies and for weak and strong environmental coupling with ${\kappa }_{p0}={10}^{-4}{\kappa }_0$ and ${\kappa }_{p0}=700{\kappa }_0$, respectively. In all panels phonons are included with ${\kappa }_{\mathrm{ph}}=0.01{\kappa }_0$. By adjusting $|\mu -{\mu }_{\mathrm{Dirac}}|$ the temperature oscillation wavelength can be tuned. Even with strong environmental coupling and significant phonon heat conductance, the quantum temperature oscillations are visible. The phonon temperature $T_{\mathrm{ph}}$ is taken to vary linearly between each electrode, and the applied temperature gradient across the nanoribbon is 50 K.

Figure 3

The power spectral density (PSD) of a slice through the edge row of the GNR shown in Fig. 2 as a function of gate potential. The temperature oscillation wavelength increases as $\mu -{\mu }_{\mathrm{Dirac}}$ is decreased, in reasonable agreement with Eq. (4), whose values are indicated by vertical blue lines. The PSD spectra are complex because of the small size of the GNR, the multimode nature of the Pt SThM, and the phonon conductance. The temperature data within 10 Å of each electrode have been neglected in the PSD spectra and ${\kappa }_{p0}={10}^{-4}{\kappa }_0$.

Figure 4

The simulated temperature profile of a graphene fragment without (left panel) and with (right panel) impurities contacted by a hot $\left(T_1=350\mathrm{K}\right)$ needle electrode (the benzene-like contact pattern is indicated with red circles) and a “cold” electrode $\left(T_2=T_0=300\mathrm{K}\right)$ bonded to the periphery of the sheet (blue circles) probed by a Pt SThM tip scanned 2.5 Å above the plane of the carbon nuclei. In these simulations, we use ${\kappa }_{p0}=700{\kappa }_0$ (extracted from experiment) and ${\kappa }_{\mathrm{ph}}=0.01{\kappa }_0$. The hot needle and periphery electrodes have per orbital coupling strengths of 1 eV and 0.1 eV, respectively. In the right panel, 0.33% boron (black circles), 0.33% nitrogen (white circles), and 0.33% vacancy impurities were included. Here $\mu -{\mu }_{\mathrm{Dirac}}=-1\text{eV}$.